Note on the normal form of a spatial stiffness matrix

There has been some recent interest in the problem of designing compliance mechanisms with a given spatial stiffness matrix. A key result that has proven useful in the design of such mechanisms is Loncaric´s normal form. When a spatial stiffness matrix is described in an appropriate coordinate frame, it will have a particularly simple structure. In this form the 3×3 off-diagonal blocks of the stiffness matrix are diagonal. It has been shown that generically, a spatial stiffness matrix can be written in normal form. For example, it is fairly well known that this is possible for any positive definite spatial stiffness matrix. In this article, it is shown that any symmetric positive semi-definite matrix can be written in normal form. As an application this result is used to design a compact parallel compliance mechanism with a prescribed positive semi-definite spatial stiffness matrix